Can someone verify whether i am correct in saying that the square root of negative one SHOULD NOT exist, make sense, or equal "i" or +/-i?

Question

I am sure you are aware of the dillema of 1=-1 when we assume √-1=i. To refresh your memory it goes like this:

1= √1= √((-1)(-1))= √-1√-1= i*i= -1.

We must realize that saying √-1=i is only half true, therefore not true. √-1= +/-i. This is because i×i=-1 and so does -i×-i. Therefore we need to change our above statement to:

1= √1= √((-1)(-1))= √-1√-1= +/-i*+/-i= (+/-)-1=(+/-)1.

But then, wait, now 1 is equal to +/-1. It is equal to both itself and its opposite all of the sudden. Doesnt this make absolutely no sense? I think √-x should equal the set of no numbers. What do all of the educated mathematicians say?

Answer 1

The "laws of exponents" need modification when dealing with complex numbers. In particular, it is not true that $\sqrt{xy} = \sqrt{x} \sqrt{y}$.

Answer 2

Let's examine your chain of equalities:

$1 =\sqrt 1 = \sqrt{(-1)(-1)} = i*i = -1$.

At this junction, it is important to make an interpretation of what $1 = \sqrt{1}$ means. What does it mean? For this statement to be well defined, the square root of any value must be unique. Do you see why? For otherwise, I could very well say the following: $1 = \sqrt 1 = -1$ !

Hence, the square root when used in this expression is a well defined function, that is to say that the square root of a number is unique. However, this is only possible if the square root is restricted to the positive numbers (which is done, precisely to avoid the situation you have brought up!). Which renders $\sqrt{-1}$ meaningless.

The moral of the story is that the square root may have the same symbol, but is a different demon when you change domains, as you have done, from the natural numbers to the whole set of integers.

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That should answer your question, but as an aside I think I should add the precise meaning.

Let $\mathbb{R}^+$ be the positive real line. Define $\sqrt{x} : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ by the rule that $(\sqrt{x})^2=x$. Well-definition is easy to check.

Now, what do we do when we have negative numbers? Worse, what do we do when we have negative powers? Read up on how the logarithm and exponential are implemented on the complex plane, because these functions are the key to every question you will have about powers of numbers.